Playlists: refactor PlaylistParser to be less mutable

There's no need to make PlaylistParser.Parse() method depend
on whether some previous code set the property BaseUri to
some value beforehand.

All current uses of the setter of the property can simply be
replaced by a constructor that receives the value for BaseUri.
By this we avoid any race condition that could be derived from
reading the object after/before setting the property. Its
getter access wasn't used at all so we can remove the property
altogether, leaving the private field for its main usage in
the Parse() method.
Playlists: tidy

Remove some unused usings, fix some indentation and missing braces.